US 8078659 B2 Abstract An apparatus for reducing a digital filter delay includes means for determining the magnitude response of a desired filter. Means form the real cepstrum of this magnitude response. Means transform the real cepstrum into a complex cepstrum of a corresponding minimum-phase filter having the same magnitude response as the desired filter. A filter applies a smoothly decaying shaping window to the complex cepstrum. Means transform the shaped complex cepstrum into an estimated minimum-phase filter.
Claims(10) 1. A method of reducing a digital filter delay, including the steps of
determining a first finite sequence of coefficients representing a real cepstrum of a magnitude response of a desired digital filter,
transforming said first sequence into a second sequence of coefficients representing a corresponding complex cepstrum,
determining an estimate of a minimum-phase digital filter corresponding to said second sequence, where said transforming step includes shaping of coefficients to be included in said second sequence by applying a smoothly decaying window function.
2. The method of
3. The method of
4. The method of
5. An apparatus for reducing a digital filter delay, wherein said apparatus is configured to perform the following:
determine a first finite sequence of coefficients representing a real cepstrum of a magnitude response of a desired digital filter,
transform said first sequence into a second sequence of coefficients representing a corresponding complex cepstrum, and
determine an estimate of a minimum-phase digital filter corresponding to said second sequence, where said apparatus is configured to transform said first sequence by applying a smoothly decaying window function to shape coefficients to be included in said second sequence.
6. The apparatus of
7. The apparatus of
8. The apparatus of
9. An echo canceller for a telecommunication system, including a non-linear processor with a serially connected echo canceling digital filter, wherein said echo canceller comprises:
an apparatus for reducing the delay of said filter, wherein said apparatus is configured to perform the following:
determine a first finite sequence of coefficients representing a real cepstrum of a magnitude response of a desired digital filter,
transform said first sequence into a second sequence of coefficients representing a corresponding complex cepstrum, and
determine an estimate of a minimum-phase digital filter corresponding to said second sequence, where said apparatus is configured to transform said first sequence by applying a smoothly decaying window function to shape coefficients to be included in said second sequence.
10. A noise suppressor including a serially connected noise suppressing digital filter, said noise suppressor comprising:
an apparatus for reducing the delay of said filter, wherein said apparatus is configured to perform the following:
determine a first finite sequence of coefficients representing a real cepstrum of a magnitude response of a desired digital filter,
transform said first sequence into a second sequence of coefficients representing a corresponding complex cepstrum, and
determine an estimate of a minimum-phase digital filter corresponding to said second sequence, where said apparatus is configured to transform said first sequence by applying a smoothly decaying window function to shape coefficients to be included in said second sequence.
Description The present invention relates to reduction of delay in digital filters based on a desired magnitude response. Requirements on a digital signal processing device to be used in a telephony network normally aim at minimizing the signal propagation delay imposed by the device, to reduce the total round trip delay which otherwise would affect the conversation quality negatively, see [1]. It is common for signal processing devices (such as noise reduction devices, echo cancellers, etc.) to apply some kind of linear filtering, serially connected to the signal path, to improve the speech signal. Two examples are: - 1. Noise reduction devices, where the common solution is to apply filtering so that the disturbing background noise is reduced while at the same time retaining the speech so as to improve intelligibility.
- 2. Echo cancellation devices, where typically a linear adaptive filter (connected in parallel with the echo path) models the echo path to enable subtraction of a replica of the echo and thus a reduction of the echo level. However, due to imperfections in the adaptive algorithms and due to time varying and possibly non-linear echo paths, residual echo is left after the subtraction. This residual echo is typically removed by a non-linear processor (NLP), which detects the presence of the residual echo and, if found, removes it (optionally replacing it with some kind of generated comfort noise). The NLP is usually implemented as a switch that either passes or suppresses the signal applied to it. However, the NLP may also be implemented as a serially connected linear filter, as described in [2].
In both these telecommunication applications it is vital that the voice enhancement devices do not add to the round trip delay. Thus, it is desirable to minimize the delay of the signal processing filters. However, at the same time it is also desirable to maintain high stop-band rejection. Achieving high stop-band rejection is particularly important in echo cancellation applications, where a serially connected filter is required to suppress residual echo in certain frequency bands (the ones where residual echo dominates over true near end signal), while passing some other bands (the ones dominated by the true near end signal). The derivative of the phase characteristic of a linear filter with respect to frequency is a measure of the time delay (i.e. group delay or envelope delay) that signal frequency components undergo in passing through the filter. A minimum-phase filter characteristic implies a minimum delay function, while a maximum-phase filter characteristic implies that the delay characteristic is also maximum. A mixed-phase system has a delay in between the two extremes. One solution to decrease the delay, which is imposed by a linear filter, is to convert it to a minimum-phase filter, which by definition has a minimal group delay through the filter. Suppose that we have a finite impulse response (FIR) filter prototype H(ω) with real coefficients. The magnitude square value of its frequency response is
Relationship (1) implies that if we replace a zero z - A. Spectral factorisation, e.g. using some approximating recursive algorithms as described in [3, 4]. However, a problem with spectral factorisation is that it is computationally demanding and that it needs an increasing number of iterations to reliably form a minimum-phase filter of a demanding filter prototype (i.e. a complicated desired magnitude response with high dynamic range) with high required stop band attenuation.
- B. Employing homomorphic filtering as described in, for example, [5]. The method essentially determines the real cepstrum of the magnitude response of the desired digital filter, transforms this into a corresponding complex cepstrum representing a minimum-phase filter, and determines the minimum-phase digital filter corresponding to the complex cepstrum. The problem with this method is that it requires long discrete Fourier transforms (DFTs) to attain high stop band rejection in the resulting minimum-phase filter and thus becomes computationally demanding.
- C. The errors imposed due to the usage of a discrete Fourier transform (DFT) in method B have been analysed in [6], and a method is proposed to reduce these errors. The method basically trusts on the usage of long DFTs together with truncation of the resulting impulse responses to minimise the effects of aliasing. This results in that the effective length of the impulse response will be half of the length of the used DFT. When using short DFTs, this leads to difficulties in obtaining the desired magnitude response of the prototype filter.
An objective of the present invention is to provide a method and apparatus for reducing the delay of a desired filter with a given magnitude response to approximately minimum-phase with shorter DFTs than in the prior art. Another objective is an echo canceller including such an apparatus. Still another objective is a noise suppressor including such an apparatus. These objectives are achieved in accordance with the attached claims. Briefly, the present invention is based on method B above, but shapes the coefficients to be included in the complex cepstrum corresponding to the minimum-phase filter by applying a smoothly decaying window function. This makes it possible to obtain estimates of minimum-phase filters from demanding magnitude responses by using short DFTs. The use of short DFTs results in a low complexity solution compared to the prior art solutions, without sacrificing the requirements on high stop band rejection. The invention, together with further objects and advantages thereof, may best be understood by making reference to the following description taken together with the accompanying drawings, in which: To better explain how the idea of the present invention is related to minimum-phase filter design, let us first reproduce some of the theory. Suppose we have a finite impulse response (FIR) filter
The system function of the filter h is given by its z-transform
Denote the logarithm of the magnitude response of the system function above as
The cepstrum, or more accurately the real cepstrum, of h is defined as the coefficients ĥ(n) of the power series expansion of Ĥ(z), i.e.
Choosing the integration contour, C, to be the unit circle, (6) transforms to
Denote the zeros of H(z) by a
Computing the logarithm of the square root of the above expression results in
Using the following power series representations
Comparing the above with (5) and identifying coefficients of the same power of z it is evident that
From (14) it may be noted that the components originating from the zeros inside the unit circle only contribute to the cepstrum at positive n and that the components originating from zeros outside the unit circle only contribute to the cepstrum at negative n. A minimum-phase filter has all its zeros inside the unit circle. Hence, to get the cepstrum corresponding to a minimum-phase filter one needs to set the cepstrum at negative n to zero. Furthermore, from (14) it may also be noted that the real cepstrum has infinite duration although the original filter h(n) has a finite length. A similar reasoning as above can be applied to the complex cepstrum (see [5]), i.e.
Comparing (16) with (14), it is evident that the complex cepstrum of a minimum-phase sequence can be restored from its real cepstrum as
In order to obtain the impulse response of the minimum-phase filter one needs to reverse the process, i.e. compute
The outlined method may be implemented by using the discrete Fourier transform (DFT) and its inverse (IDFT), as described in [5] and illustrated in Here M must be sufficiently large to avoid aliasing (M=1024 in the example given in [5]). Step S
Note that due to the periodicity assumed by the IDFT, the cepstral components with negative n are (by convention) located between M/2+1 and M−1 in equation (23). Finally step S The resulting filter h When computing the minimum-phase impulse response, a problem arises in practice due to the fact that it is necessary to replace the infinite coefficient sequence ĥ In step S In the embodiment described above a Kaiser window has been used for shaping the sequence of coefficients ĥ From Shannon's sampling theorem it is known that replacement of the integrals in Fourier transforms by sums can be done without error if the spectrum of the signal is zero for all frequencies above half the sampling frequency. In the problem at hand, the logarithm of the system function Ĥ(ω)=ln|H(ω)| must be sampled before computing the inverse Fourier transform. Hence, it should be ensured that a condition similar to that of the sampling theorem holds for ln|H(ω)|. Typically in echo cancellation and noise reduction applications, the dynamically updated desired magnitude response H(ω
As can be seen in From In the description above a Hanning window has been described (due to its simple kernel) for smoothing the magnitude response. However, other smoothing windows, such as Hamming, Blackman, Bartlett, Kaiser windows, are also feasible. As an alternative, the filtering of input signal x(n) may also be performed in the time domain, provided that the minimum-phase filter from block The different blocks of the echo canceller of Although the present invention has been described in detail with reference to echo cancellation, the same principles may be used in other applications, for example noise reduction in telecommunication or audio systems. As is described in [4] this application may be based on a serially connected noise reduction filter with a desired magnitude response, the delay of which should be as small as possible. The essential difference is that instead of determining the desired magnitude response from an estimate {circumflex over (Φ)} The present invention proposes a method for design of minimum-phase filters from a magnitude frequency response (with arbitrary phase) using only short DFTs. This results in a low complexity method, which outperforms all other known methods in terms of complexity and resulting stop band attenuation of the resulting minimum-phase filters. The method is particularly suitable for environments where a multitude of identical channels have to be processed (e.g. noise reduction, echo cancelling devices) at a minimum cost of computational operations and heat dissipation. The method allows for processing of more channels simultaneously than would have been possible using state of the art methods, given a limited amount of computational power. Alternatively, the reduced complexity provided by the present invention may be used to reduce heat dissipation and/or improve battery utilization. It will be understood by those skilled in the art that various modifications and changes may be made to the present invention without departure from the scope thereof, which is defined by the appended claims.
- [1] ITU-T G.131, Talker echo and its control
- [2] U.S. Pat. No. 658,107 B1
- [3] Kucera V., Discrete linear control, John Wiley & Sons, 1979
- [4] WO 01/18960 A1
- [5] A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing, Prentice-Hall, Inter. Ed., 1989, Chapter 12
- [6] Bysted, T. K., Aliasing in the complex cepstrum of linear-phase signals, International Conference on Information, Communication and Signal Processing, 1997
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